Lexicographic decomposition of preferences

The Ordinal Lexicographic Model, based on the lexicographic sum of relations, provides an explanation to intransitivities in preferences as shifts in choice criteria. The lexicographic sum R⊕R′ between two preferences R and R′ on a same finite set X is defined as follows: x is preferred to y with respect to R⊕R′ when x is preferred to y with respect to R, or there is a tie between x and y with respect to R but x is preferred to y with respect to R′. If a preference R can be written as R=R1⊕R2⊕…⊕Rk,(R1,R2,…,Rk) is said to be a lexicographic decomposition (LD) of R. It means that a first criterion, expressed by R1, explains a part of R, a second criterion, expressed by R2, explains a part of R which is not explained by R1, and so on. Each relation Ri of the LD can be interpreted as a point of view, and the number k is the number of shifts in point of view in the decomposition. It is usually required from the relations of the LD to fulfil some structural properties, for instance to be partial orders. When a LD is possible, the usual question consists in computing, for any preference R, the minimum number d(R), called the lexicographic dimension of R, of relations involves in the LD in order to explain R. The aim of this paper is to provide some properties of the lexicographic sum ⊕.